Definition of shear force at a section of a beam For a simply supported (or any) beam, the shear force at a given cross-section is equal to the:

Introduction / Context:
Shear force and bending moment diagrams are constructed from equilibrium of a cut section. A precise definition of shear force at a section is essential before solving for reactions, internal forces, and drawing diagrams.

Given Data / Assumptions:

• Beam subjected to transverse loading (point loads, UDLs, etc.).
• Statics: equilibrium of a free-body diagram (FBD) of part of the beam.
• Sign convention for shear: typically upward on left face positive (or another consistent convention).

Concept / Approach:

At a chosen section, we isolate one part of the beam. The internal shear force V at that section balances the algebraic sum of all external transverse forces on the isolated part. Therefore, V equals the algebraic sum (with sign) of transverse forces acting on either the left or the right segment of the cut, not the entire beam.

Step-by-Step Solution:

Verification / Alternative check:

Consistency check: Shear force diagram slope equals negative of load intensity (dV/dx = −w), derived from this definition and equilibrium, reinforcing the sign-aware summation concept.

Why Other Options Are Wrong:

• (a) and (e) are deformation measures, not force definitions.
• (b) takes forces from both sides and along all directions, which is not how shear is defined at a section.
• (c) omits algebraic sign, losing directionality necessary for equilibrium.

Common Pitfalls:

• Summing forces on both sides of the cut, which trivially gives zero and is meaningless for finding internal forces.

Final Answer:

Algebraic sum of the transverse forces on one side of the section.